The mean function \(m\)  can be interpreted as the mean at any input point and the covariance function \(k\)  as the similarity between values of \(f(\rho )\)  on different inputs \(\rho \) . The covariance function is discussed in Section REF  and the mean function is assumed to be zero. Note that this is not necessary, see [1]}. Training data for a GP are defined by sampling the function \(f\)  on inputs and measuring the (noisy) training outputs \(y\)
\(\begin{aligned}y &=f(\rho )+\epsilon , \\\text{ where } \epsilon & \sim \mathcal {N}\left(0, \sigma _{\epsilon }^{2} I\right),\end{aligned}\) 
